THE CHOICE BETWEEN MULTIPLICATIVE AND ADDITIVE OUTPUT UNCERTAINTY

Transcription

1 THE CHOICE BETWEEN MULTIPLICATIVE AND ADDITIVE OUTPUT UNCERTAINTY Moawia Alghalith University of St. Andrews and Ardeshir J. Dalal Northern Illinois University Abstract When modeling output uncertainty, the multiplicative specification is consistently chosen over the additive form, despite the latter being arguably intuitively more obvious. The rationale for this seems to be that when production risk is the only source of uncertainty, additive uncertainty does not reduce output below the certainty level, while multiplicative uncertainty does. We show that, regardless of the specification of output uncertainty, if hedging is absent and there is simultaneous price and output uncertainty, output is always lower than the situation in which one or both sources of uncertainty are absent. Thus, both models yield qualitatively identical results, i.e., adding a source of uncertainty reduces expected output. Therefore, additive uncertainty is indeed a reasonable a priori method of modeling production uncertainty. Running Head: Additive and Multiplicative Uncertainty JEL Classification: D21, D81. Key Words: Multiplicative output uncertainty, additive output uncertainty, price uncertainty. Mailing Address: Ardeshir J. Dalal Department of Economics Northern Illinois University DeKalb, IL Phone No.: (815) Fax: (630) adalal@niu.edu

2 THE CHOICE BETWEEN MULTIPLICATIVE AND ADDITIVE OUTPUT UNCERTAINTY 1. Introduction In the literature on the theory of the firm under uncertainty one can find a variety of alternative specifications of production uncertainty, since no single, standard method of modeling production uncertainty exists. Letting q denote random output, some of the methods which have been used include introducing a random variable into the production function so that q = q (x, ε), where x is a single input and ε is a random production shock (see Viaene and Zilcha (1998) and MacMinn and Holtman (1983)), or including two inputs and attaching a multiplicative random component to each so that q = f (uk, vl), where u and v are random (see Ratti and Ullah (1976)). These approaches cannot readily be generalized to include multiple inputs, and therefore have limited usefulness in providing empirically testable hypotheses. Formulations which make it possible to include multiple inputs typically introduce a cost function into the analysis and specify random output as q = vy where v is random with E [v] =1and y is non-random (e.g., Lapan and Moschini (1994), Grant (1985)). In fact such multiplicative uncertainty seems to be the most commonly used method of modeling production uncertainty. 1 Additive uncertainty in which q = y +θη where E [η] = 0and V ar(η) = 1, so that E [q] = y and Var(q) =θ 2 does not appear to be the formulation of choice. This is rather surprising, because it is, of course an obvious and perfectly general formulation which can be used for any random variable with a well-defined mean and standard deviation. 2 Thefactthatitisnotused is probably because existing analyses indicate that the implications of additive production uncertainty are no different from those which are obtained with production certainty (expected) output in the two different scenarios remains unchanged. In addition to being intuitively unappealing (after all one does expect the presence of uncertainty to cause a reduction in the optimal level of output), this seems to rule out additive uncertainty as a source of any fresh insights. In contrast, multiplicative uncertainty does imply a reduction in (expected) output. It is straightforward to establish these results if output uncertainty is the only source of uncertainty in the model, but they have also been established by Honda (1983) in a model which includes both price and output uncertainty. The inclusion of simultaneous price and output uncertainty is a desirable generalization, but Honda s framework is still fairly restrictive since he includes only 1 In addition to the authors already cited, it has been used, for instance, by Batra (1974, 1975), Kemp (1976) and Britto (1980). 2 For any random variable q with mean y and standard deviation θ, one can always define the random variable η as η = q y, where E [η] =0and Var(η) =1. θ 1

3 one input and assumes the existence of a futures market in which the firm has the opportunity to hedge at least part of its output. The drawbacks of a single input model are obvious, and the inclusion of hedging opportunities for the firm restricts the relevance of the analysis to commodities for which forward or future markets exist. This limits the applicability of Honda s results since even for agricultural products (where hedging opportunities are most prevalent) the number of commodities for which futures markets exist is only a small fraction of the total number of commodities. 3 The implications of additive production uncertainty have never been explored in a model which includes both price and output uncertainty and in which hedging opportunities do not exist. Since such a model appears to be relevant for the majority of commodities, such an exploration would seem to be worthwhile on a priori grounds alone. Moreover, if the implications of the two alternative specifications are similar, then the additive formulation provides a simple and general specification which could be useful in both theoretical and empirical work; if the implications are different, then it is a fortiori worth investigating the additive specification. 4 In this paper we extend previous results in the literature by investigating the impact on optimal (expected) output of the two alternative formulations for production uncertainty in a multiple input framework, when hedging opportunities are absent. For each specification we compare the optimal output in the presence of both price and output uncertainty with the optimal output when there is (a) complete (price and output) certainty, (b) price certainty and output uncertainty, and (c) price uncertainty and output certainty. We find that when price and output are both uncertain, regardless of whether output uncertainty is additive or multiplicative, optimal (expected) output is lower than in any of the cases (a), (b) or (c). The results suggest that in a model which includes both price and production uncertainty, additive output uncertainty should not be disregarded on a priori grounds, and that the choice between the two specifications is a non-trivial one. 2. The Model Acompetitivefirm selling a single output faces an uncertain output price given by p = p+σε, where ε is random with E [ε] =0and Var(ε) =1, so that E [p] = p and Var(p) =σ 2. The level of output realized at the end of the production process is not known ex ante, so that output 3 In the North American Industry Classification System (NAICS) 34 categories of agricultural goods are identified at the 5-digit level, and many of these have multiple components. For example, in the Vegetable and Melon Farming category (11121), one 6-digit subdivision alone (Other Vegetables) lists 69 different commodities, (All Other Grains) and (Oilseeds (except soybeans)) each include 8 items, etc. However, the Wall Street Journal lists futures markets for only 15 agricultural commodities. Detailed information on the NAICS is readily available at 4 Results obtained by Alghalith (2001) indicate that the two alternative specifications do indeed yield nontrivially differing implications. 2

4 is random and is denoted by q. We consider two alternative specifications of q. If uncertainty enters additively, then q = y + θη, whereη is random with E [η] =0and Var(η) =1. However, with multiplicative uncertainty, q = vy where v is random and E [v] =1. Thus, in both cases E [q] =y. Costsareknownwithcertaintyandaregivenbyacostfunction,c (y, w) which displays positive and increasing marginal costs so that c y (y, w) > 0 and c yy (y, w) > 0. While y represents expected output it is also the level of output which would prevail in the absence of any random shocks to output. The firm may be thought of as having y as its target level of output, and committing inputs which would generate this level in the absence of any random shocks. The cost function is then the minimum cost of producing any arbitrary output level y given the input price vector w. We assume that p and q are independently distributed for two reasons. First, this assumption may be a reasonable approximation of reality since price risk can be viewed as a market phenomenon caused by random changes in aggregate supply or demand, while output risk is firm-specific, or local, and is due to factors such as natural catastrophes, unfavorable weather, disease or insect damage, etc. so that Losq (1982, p. 69) argues that this might constitute a reasonable approximation for a number of firms. 5 Second, the result that multiplicative output uncertainty reduces optimal output has been derived only for this case, and if independence is not assumed then the result is indeterminate (see Honda, 1983). Our purpose here is to compare the output effects of additive and multiplicative output uncertainty when determinate results can be obtained, and this necessitates making the independence assumption. This assumption implies E [pq] = py, E [pη] =0, E [pv] = p. Profit isπ = pq c (y, w). The firm is risk averse and seeks to maximize the expected utility of profit. It therefore seeks to solve the problem maxe [U (π)] = E [U (pq c (y, w))]. y The first order condition for a maximum is E [U 0 (π )(pδ c y (y, w))] = 0, (1) where y is the optimal level of y, π is the corresponding level of π, δ = 1 for additive uncertainty and δ = v for multiplicative uncertainty,. Note that in either case E [δ] =1, and 5 However, see Grant (1985) for a contrasting opinion. 3

5 E [pδ] = p. 3. Certainty Comparisons An alternative, equivalent, way of writing (1) is ( p c y (y, w)) E [U 0 (π )] + Cov (U 0 (π ),pδ) =0. (2) Since Cov (U 0 (π ),pδ) < 0, (2) implies p >c y (y, w), regardless of the value of δ. 6 Two results may now be stated immediately Proposition 1. For both additive and multiplicative output uncertainty, optimal expected output in the presence of price and output uncertainty is less than the optimal output under complete certainty. 7 Proof. With no uncertainty in the model, ε 0, and either η 0 or v 1. Thus p p and q y. The optimal output with no uncertainty, ŷ, is determined by the equation p = c y (ŷ). When both price and output are uncertain, (2) holds and p >c y (y ). Hence c y (ŷ) >c y (y ). Since c yy (y) > 0 this clearly implies y < ŷ. 8 Proposition 2. Suppose output uncertainty takes the additive form. When both price and output uncertainty exist, the optimal expected output is less than when output is uncertain but price is certain. Proof. Let p p with certainty and δ =1.Then(1) implies ( p c y (ŷ)) E [U 0 (ˆπ)] = 0, where hats represent optimal values. Hence p = c y (ŷ). 9 However, with both price and output uncertainty, as already established, p >c y (y ). Hence c y (ŷ) >c y (y ) and ŷ>y. 6 Suppose δ =1. Then Cov (U 0 (π ),pδ) =Cov (U 0 (π ),p), and the independence assumption implies that π / p >0. If δ = v, then Cov (U 0 (π ),pδ) =Cov (U 0 (π ),pv), and clearly π / (pv) > 0. The fact that U 00 < 0 then implies that Cov (U 0 (π ),pδ) < 0 foreithervalueofδ. 7 Grant (1985) establishes this result for multiplicative output uncertainty but not for additive uncertainty. 8 Note that in order for the certainty output to be determinate we must have c yy (y) > 0. In order to simplify the notation, in this proof and in the remainder of the paper, we do not show explicitly the dependence of c, c y, or c yy on w. 9 Note that in the absence of price uncertainty, the optimal (expected) output with additive output uncertainty is the same as that of the complete certainty case. This is the result obtained by Honda (1983). 4

6 Proposition 3. Suppose output uncertainty takes the multiplicative form. When both price and output uncertainty exist, the optimal expected output is less than when output is uncertain but price is certain. 10 Proof. Define the sets A and A such that A = {v pv c y (y ) 0}, A = {v pv c y (y ) < 0}. For any v A and v 0 A, clearlyv>v 0. Hence, since p 0, we must have pvy c (y ) pv 0 y c (y ); v A, v 0 A; p. Since U 00 < 0, the inequality above implies U 0 (pvy c (y )) U 0 (pv 0 y c (y )) ; v A, v 0 A; p. Therefore, S max v A U 0 (pvy c (y )) I min v A U 0 (pvy c (y )). 10 Viane and Zilcha (1998) prove that output is lower with simultaneous price and output uncertainty than with output uncertainty alone. They model output uncertainty by a production function Q (X, ε), where X is a single input and ε is random. Apart from the lack of generality involved in assuming only one input, their proof is also based on the assumptions that U 0 (π) is convex and πu 0 (π) is concave. The first implies U 000 (π) 0 and the second implies U 000 (π) π +2U 00 (π) 0. These assumptions place unnecessary restrictions on the utility function, and thus lessen the generality of their proof. In contrast, our proofs hold for multiple inputs and for all strictly concave utility functions. 5

7 Thus, E p [S] U 0 (E p [π ]) E p [I] U 0 (E p [π ]), where E p denotes the expectation with respect to p for given v. Since S and I are both positive, there must exist a positive constant t such that E p [S] U 0 (E p [π ]) t E p [I] U 0 (E p [π ]), so that tu 0 (E p [π ]) E p [S] E p [U 0 (pvy c (y ))],v A (3) where the last inequality holds since S is a maximum. Now (3) implies ( pv c y (y )) tu 0 (E p [π ]) ( pv c y (y )) E p [U 0 (pvy c (y ))],v A (4) since pv c y (y ) 0 for v A. Similarly, tu 0 (E p [π ]) E p [I] E p [U 0 (pvy c (y ))],v A, (5) since I is a minimum, and then (5) implies ( pv c y (y )) tu 0 (E p [π ]) ( pv c y (y )) E p [U 0 (pvy c (y ))],v A, (6) since pv c y (y ) < 0 for v A. The inequalities in (4) and (6) thus hold for all values of v, and taking expectations with 6

8 respect to v yields te [( pv c y (y )) U 0 ( pvy c (y ))] E [( pv c y (y )) U 0 (π )]. (7) Now rewrite (1) letting δ = v and p = p + σε, to get E [U 0 (π )( pv c y (y ))] + σe [U 0 (π) vε] =0, but σe [U 0 (π ) vε] =Cov (U 0 (π ) v, p) < 0. Therefore E [U 0 (π )( pv c y (y ))] > 0, and then (7) implies E [( pv c y (y )) U 0 ( pvy c (y ))] > 0. (8) If p = p with certainty, the first order condition for a maximum is E [( pv c y (ŷ)) U 0 ( pvŷ c (ŷ))] = 0, (9) where ŷ is the optimal output when price is known with certainty. Now consider E [U (π 1 )] = E [U ( pvy c (y))], and note that E [U (π 1 )] is strictly concave in y, and the left hand sides of (8) and (9) are expressions for E[U(π 1)] evaluated at y and ŷ y respectively. The former derivative is on the upward sloping part of E [U (π 1 )], while the latter is at the maximum. Hence y < ŷ. Proposition 4. For additive (multiplicative) output uncertainty, optimal expected output in the presence of price and output uncertainty is no greater than (less than) the optimal output obtained when price is uncertain but output is not. Proof. Define the sets A and A such that A {p p c y (y ) 0} 7

9 A {p p c y (y ) < 0}. For any p A and p 0 A, clearly p>p 0. Hence, we must have pq c (y ) p 0 q c (y ); p A, p 0 A, where q is the value of q corresponding to y. Since U 00 < 0, the inequality above implies U 0 (pq c (y )) U 0 (p 0 q c (y )) ; p A, p 0 A. Therefore, S max p A U 0 (pq c (y )) I min p A U 0 (pq c (y )) Thus, E q [S] U 0 (E q [π ]) E q [I] U 0 (E q [π ]), (10) where E q denotes the expectation with respect to q for given p. Proceeding in a manner exactly parallel to that used in the proof of Proposition 3, we can show that (10) implies (p c y (y )) tu 0 (E q [π ]) (p c y (y )) E q [U 0 (pq c (y ))], p where t is a positive constant. Taking expectations of both sides with respect to p yields =0if δ =1 te [(p c y (y )) U 0 (py c (y ))] E [(p c y (y )) U 0 (π )] > 0 if δ = v, (11) 8

10 The last equality and inequality in (11) hold because (1) can be rewritten as Cov (U 0 (π ) p, δ)+e [U 0 (π )(p c y (y ))] = 0. The results in (11) follow immediately, since Cov (U 0 (π ) p, δ) < 0 for δ = v, and Cov (U 0 (π ) p, δ) = 0 for δ =1. Since t>0, (11) implies 0 if δ =1 E [(p c y (y )) U 0 (py c (y ))] > 0 if δ = v. (12) If output uncertainty does not exist then q y and the first-order condition for a maximum is E [(p c y (ŷ)) U 0 (pŷ c (ŷ))] = 0, (13) where ŷ is the optimal output when output is not subject to uncertainty. Now let E [U (π 2 )] = E [U (py c (y))], and note that E [U (π 2 )] is strictly concave in y. The left hand sides of (12) and (13) are expressions for E[U(π 2)] evaluated at y and ŷ respectively. y The former derivative is either on the upward sloping part of E [U (π 2 )] or at the maximum (if (12) holds with equality), while the latter is at the maximum. Hence y 6 ŷ, with the strict inequality holding for δ = v. 4. Conclusion In the existing literature on output uncertainty, the multiplicative specification is consistently chosen in preference to the additive form, despite the latter being arguably intuitively more obvious.therationaleforthisseemstobethatwhenproductionuncertaintyistheonlysource of uncertainty, additive uncertainty does not reduce output below the certainty level, while multiplicative uncertainty does. We show that the results in the two cases no longer diverge so sharply when the framework of the model is modified to include price uncertainty as well. We have shown that regardless of the specification of output uncertainty, if hedging is absent and there is simultaneous price and output uncertainty, output is always lower than the situation in which one or both sources of uncertainty are absent. Further, we obtain these results within 9

11 the framework of a multiple input production process. While the magnitude of the optimal expected output may differ in the two models, both yield qualitatively identical results, i.e., adding a source of uncertainty reduces expected output. Since both cases yield the intuitively appealing result that adding a source of uncertainty reduces the optimal expected output, intuition alone does not provide a criterion for choosing between the two alternative specifications of output uncertainty. If one method is superior to the other, this would need to be determined by using some alternative criteria. Our results indicate that in contrast to the general practice in the literature, additive uncertainty provides a reasonable a priori method of modeling production uncertainty. 10

12 REFERENCES Batra, R. Resource Allocation in a General Equilibrium Model of Production Under Uncertainty, Journal of Economic Theory, 1974, 8, Batra, R. The Pure Theory of International Trade Under Uncertainty. John Wiley, New York, Britto, R. Resource Allocation in a Simple Two-Sector Model with Production Risk, Economic Journal, 1980, 90, Grant, D. Theory of the Firm with Joint Price and Output Risk and a Forward Market. American Journal of Agricultural Economics, 1985, 67, Honda, Y. Production Uncertainty and the Input Decision of the Competitive Firm Facing the Futures Market. Economics Letters, 1983, 11, Kemp, M. Some General Equilibrium Implications of Technological Uncertainty, in Three Topics in the Theory of International Trade, North-Holland, Amsterdam, Lapan, H. and G. Moschini. Futures Hedging Under Price, Basis, and Production Risk. American Journal of Agricultural Economics, 1994, 76, Losq, E. Hedging with Price and Output Uncertainty. Economics Letters, 1982, 10, MacMinn, R. and Holtman, A. Technological Uncertainty and the Theory of the Firm. Southern Economic Journal, 1983, 50, Ratti, R. and Ullah, A. Uncertainty in Production and the Competitive Firm. Southern Economic Journal, 1976, 42, Viaene, J. and Zilcha, I. The Behavior of Competitive Exporting Firms under Multiple Uncertainty. International Economic Review, 1998, 39,